And with this culmination our Western mathematic, having exhausted every inward possibility and fulfilled its destiny as the _copy and purest expression of the idea of the Faustian soul_, closes its development in the same way as the mathematic of the Classical Culture concluded in the third century. Both those sciences (the only ones of which the organic structure can even to-day be examined historically) arose out of a wholly new idea of number, in the one case Pythagoras’s, in the other Descartes’. Both, expanding in all beauty, reached their maturity one hundred years later; and both, after flourishing for three centuries, completed the structure of their ideas at the same moment as the Cultures to which they respectively belonged passed over into the phase of megalopolitan Civilization. The deep significance of this interdependence will be made clear in due course. It is enough for the moment that for us the time of the _great_ mathematicians is past. Our tasks to-day are those of preserving, rounding off, refining, selection— in place of big dynamic creation, the same clever detail-work which characterized the Alexandrian mathematic of late Hellenism. A historical paradigm will make this clearer. column 1 column two ------------------------------------------------------------------------ CHAPTER III THE PROBLEM OF WORLD-HISTORY I PHYSIOGNOMIC AND SYSTEMATIC ------------------------------------------------------------------------ CHAPTER III THE PROBLEM OF WORLD-HISTORY I PHYSIOGNOMIC AND SYSTEMATIC I Now, at last, it is possible to take the decisive step of sketching an image of history that is independent of the accident of standpoint, of the period in which this or that observer lives—independent too of the personality of the observer himself, who as an interested member of his own Culture is tempted, by its religious, intellectual, political and social tendencies, to order the material of history according to a perspective that is limited as to both space and time, and to fashion arbitrary forms into which the superficies of history can be forced but which are entirely alien to its inner content. What has been missing, till now, is _detachment_ from the objects considered (die Distanz vom Gegenstande). In respect of Nature, this detachment has long ago been attained, though of course it was relatively easy of attainment, since the physicist can obviously systematize the mechanical-causal picture of his world as impersonally as though he himself did not exist in it.